A213158 Positive integers of the form (x+y+z)*x*y*z (x,y,z positive integers).

3, 8, 15, 20, 24, 35, 36, 48, 56, 63, 80, 84, 96, 99, 108, 120, 128, 135, 140, 143, 144, 168, 176, 180, 195, 200, 216, 224, 231, 240, 243, 255, 260, 264, 275, 288, 300, 308, 320, 323, 336, 351, 360, 384, 396, 399, 416, 420, 440, 455, 468, 476, 483, 495, 504

Offset

1,1

Comments

Square terms are 36, 144, 576,... and the corresponding square roots are 6, 12, 24,... i.e. sequence A188158 (integer areas of primitive integer triangles).

Positive integers of the form (a^2-b^2)*(b^2-c^2) with integers a>b>c>=0. - Michael Somos, May 18 2013

References

R. D. Carmichael, Diophantine Analysis, Wiley, 1915, p. 9.

Links

Table of n, a(n) for n=1..55.

Example

a(21)=144 for x=1, y=4 and z=4 then the triangle sides are x+y = 5, z+x = 5 and y+z = 8, hence half-perimeter = p = x+y+z = 9 and Heron's formula is checked: area = sqrt(p*(p-5)*(p-5)*(p-8)) = sqrt(144) = 12.
36 = (4^2-2^2) * (2^2-1^2). 63 = (5^2-2^2) * (2^2-1^2) = (5^2-4^2) * (4^2-2^2)= (8^2-1^2) * (1^2-0^2). - Michael Somos, May 19 2013

Mathematica

nmax = 25; mx = nmax (nmax + 2); Union[Reap[Do[a = (x + y + z)*x*y*z; If[a <= mx, Sow[a]], {x, 1, nmax}, {y, x, nmax}, {z, y, nmax}]][[2, 1]]]

Crossrefs

Cf. A188158.

Sequence in context: A310311 A310312 A310313 * A228880 A310314 A181027

Adjacent sequences: A213155 A213156 A213157 * A213159 A213160 A213161

Keyword

nonn

Author

Jean-François Alcover, Jun 06 2012

Status

approved